Admissibility for quasiregular representations of exponential solvable Lie groups

Vignon Oussa¹

¹ Bridgewater State University Bridgewater, MA 02325, U.S.A.

Colloquium Mathematicum, Tome 131 (2013) no. 2, pp. 241-264.

Résumé

Let $N$ be a simply connected, connected non-commutative nilpotent Lie group with Lie algebra $\mathfrak {n}$ of dimension $n.$ Let $H$ be a subgroup of the automorphism group of $N.$ Assume that $H$ is a commutative, simply connected, connected Lie group with Lie algebra $\mathfrak {h}.$ Furthermore, assume that the linear adjoint action of $\mathfrak {h}$ on $\mathfrak {n}$ is diagonalizable with non-purely imaginary eigenvalues. Let $\tau =\mathop {\rm Ind} _{H}^{N\rtimes H} 1$. We obtain an explicit direct integral decomposition for $\tau $, including a description of the spectrum as a submanifold of $(\mathfrak {n}+\mathfrak {h})^{\ast }$, and a formula for the multiplicity function of the unitary irreducible representations occurring in the direct integral. Finally, we completely settle the admissibility question for $\tau $. In fact, we show that if $G=N\rtimes H$ is unimodular, then $\tau $ is never admissible, and if $G$ is non-unimodular, then $\tau $ is admissible if and only if the intersection of $H$ and the center of $G$ is equal to the identity of the group. The motivation of this work is to contribute to the general theory of admissibility, and also to shed some light on the existence of continuous wavelets on non-commutative connected nilpotent Lie groups.

Détail

DOI : 10.4064/cm131-2-7

Keywords: simply connected connected non commutative nilpotent lie group lie algebra mathfrak dimension subgroup automorphism group assume commutative simply connected connected lie group lie algebra mathfrak furthermore assume linear adjoint action mathfrak mathfrak diagonalizable non purely imaginary eigenvalues tau mathop ind rtimes obtain explicit direct integral decomposition tau including description spectrum submanifold mathfrak mathfrak ast formula multiplicity function unitary irreducible representations occurring direct integral finally completely settle admissibility question tau rtimes unimodular tau never admissible non unimodular tau admissible only intersection center equal identity group motivation work contribute general theory admissibility shed light existence continuous wavelets non commutative connected nilpotent lie groups